Question

Find the dimension of each of the following vector spaces. (a) The vector space of all diagonal n x n matrices. (b) The vector space of all symmetric n x n matrices. (c) The vector space of all upper triangular n x n matrices.

Answer 1

The dimension of each vector space is determined by the number of independent entries in the matrices.

Step-by-step explanation:

Dimension of each vector space:

1. The vector space of all diagonal n x n matrices has dimension n.
2. The vector space of all symmetric n x n matrices has dimension n(n+1)/2.
3. The vector space of all upper triangular n x n matrices has dimension n(n+1)/2.

In each case, the dimension is determined by the number of independent entries in the matrices.